Representation of Qubit States by Probability Vectors

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

Any qubit state is associated with a six-dimensional probability vector with components , where is the spin projection and defines a direction of spin projection measurement, . The ends of the vectors are on the sphere which is illustrated in the top-left corner. In general, is a probability distribution function of two discrete variables and , and determines a point on the five-simplex. If the directions are chosen with equal probability, then for all . In that case, a one-to-one correspondence can be established between all probability vectors and all points inside a cube , , which is illustrated in the top-right corner. In other words, any quantum state is associated with a probability vector of the form



where is a cube in of side .

The density operator is expressed through the probabilities by


where is the identity operator, are Pauli operators, and the vectors form a dual basis with respect to the vectors :

, , .

Non-negativity of the density operator is a necessary condition that leads to constraints on the probabilities . Using Sylvester's criterion, one obtains restrictions of the first and the second order (the blue and red surfaces inside the cube, respectively). In the probability space, the set of quantum states is an ellipsoid located between two planes. The set of qubit states is depicted in the top-right corner for any choice of directions .

The errors of experimentally measured probabilities result in the reconstruction procedure above being erroneous. The error bar is directly proportional to the condition number of the Gram matrix , which is the ratio of the absolute values of the maximum to the minimum eigenvalue. The behavior of the condition number is shown at the bottom.


Contributed by: Sergey Filippov and Vladimir I. Man'ko (February 2010)
Based on a program by: S. M. Blinder
Open content licensed under CC BY-NC-SA



Representation of spin states by finite dimensional probability vectors is considered in

S. Filippov and V. Man'ko, arXiv, "Inverse Spin-s Portrait and Representation of Qudit States by Single Probability Vectors," 2010.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.